Some Properties of Continuous K-frames in Hilbert Spaces

Sahand Communications in Mathematical Analysis (SCMA) Vol. 15 No. 1 (2019), 169-187 http://scma.maragheh.ac.ir DOI: 10.22130/scma.2018.85866.432 Some Properties of Continuous K-frames in Hilbert Spaces Gholamreza Rahimlou1 , Reza Ahmadi2∗ , Mohammad Ali Jafarizadeh3 , and Susan Nami4 Abstract. The theory of continuous frames in Hilbert spaces is extended, by using the concepts of measure spaces, in order to get the results of a new application of operator theory. The K-frames were introduced by Găvruta (2012) for Hilbert spaces to study atomic systems with respect to a bounded linear operator. Due to the structure of K-frames, there are many differences between K-frames and standard frames. K-frames, which are a generalization of frames, allow us in a stable way, to reconstruct elements from the range of a bounded linear operator in a Hilbert space. In this paper, we get some new results on the continuous K-frames or briefly cK-frames, namely some operators preserving and some identities for cK-frames. Also, the stability of these frames are discussed. 1. Introduction Nowadays, frames are used in some various branches of science and engineering. Among them are signal processing, image processing, data compression and sampling in sampling theory (see [2, 3, 5, 10]). Frames were introduced by Duffin and Schaeffer in the context of Non-harmonic Fourier series [7]. They were intended as an alternative to the orthonormal or Riesz bases in Hilbert spaces. Much of the abstract theory of frames is elegantly laid out in that paper. A frame is a family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into an expansion of the frame elements (see [4, 8, 9, 11, 14]). 2010 Mathematics Subject Classification. Primary 42C15; Secondary 42C40, 41A58. Key words and phrases. K-frame, c-frame, cK-frame, Local cK-atoms. Received: 09 May 2018, Accepted: 15 October 2018. ∗ Corresponding author. 169 170 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI The theory of continuous frames in Hilbert spaces, using the concepts of measurment spaces, in order to get the results of a new application of operator theory is extended. The concept of a generalization of frames to an indexed family by some locally compact spaces endowed with a Radon measure was proposed by G. Kaiser [10] and independently by Ali, Antoine and Gazeau [1]. These frames are known as the continuous frames. In continuous K-frames, the lower bound of the frame is replaced by the norm of a bounded operator on a Hilbert space. This changes the overall structure of the frame and gives new results in terms of combining operators and frame perturbation. This paper consists of four sections. We review the foundation for the theory of continuous frames in Hilbert spaces in Section 1. The necessary tools to construct a contiuous frame will be provided. Also the structure of continuous K-frames is expressed. In Section 2, the operators that preserve continuous K-frames are discussed. In Section 3, we present some useful identities and inequalities for those frames. Finally, we study the perturbation of continuous K-frames and the lower bound of frames by using a new technique for getting perturbation of continuous K-frames in Section 4. Throughout this paper, H, H0 , H1 and H2 are Hilbert spaces, (H)1 is the closed unit ball in H. (X, µ) is a σ-finite measure space, L(H0 , H) is the set of all linear mappings of H0 to H and B(H0 , H) is the set of all bounded linear mappings. Instead of B (H, H), we simply write B (H). Also for brevity, continuous K-frame is denoted by cK-frame. Definition 1.1. Let {fn } ⊆ H. We say that the sequence {fn } is a frame for H if there exist constants A, B > 0 such that ∑ A ∥h∥2 ≤ |⟨h, fn ⟩|2 ≤ B ∥h∥2 , h ∈ H. n Definition 1.2. Let F : X → H be a weakly measurable mapping (i.e., for all h ∈ H, the mapping x 7→ ⟨F (x), h⟩ is measurable). Then F is called a c-frame for H if there exist 0 ≤ A ≤ B < ∞ such that for all h ∈ H, ∫ 2 A ∥h∥ ≤ |⟨F (x), h⟩|2 dµ ≤ B ∥h∥2 . X The constants A and B are called c-frame bounds. If A, B can be chosen so that A = B, we call this c-frame an A-tight frame, and if A = B = 1 it is called a c-Parseval frame. If we only have the upper bound, we call f a c-Bessel mapping for H. The representation space employed in this setting is L2 (X, H) = {φ : X → H|φ is measurable and ∥φ∥2 < ∞} , SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 171 (∫ 2 )1 2 where ∥φ∥2 = X ∥φ(x)∥ dµ . For each F, G ∈ L2 (X, H) , the mapping x → ⟨F (x) , G (x)⟩ of X to C is measurable, and it can be proved that L2 (X, H) is a Hilbert spase with the inner product defined by ∫ ⟨F (x) , G (x)⟩ dµ. ⟨F, G⟩L2 = X We shall write L2 (X) when H = C. Theorem 1.3 ([8]). Let F : X → H be a c-Bessel mapping for H, and U ∈ B (H, H0 ). Then U F : X → H0 is a c-Bessel mapping for H0 with U TF = TU F . Theorem 1.4 ([6]). Suppose the H, H1 and H2 are Hilbert spaces, L1 ∈ B (H1 , H) and L2 ∈ B (H2 , H). Then the following assertions are equivalent: (i) R (L1 ) ⊂ R (L2 ), (ii) ∃λ ≥ 0, such that L1 L∗1 ≤ λL2 L∗2 , (iii) There exists X ∈ B (H1 , H2 ) such that L1 = L2 X. Definition 1.5. Let K ∈ B (H0 , H), and {fn } ⊆ H. We say that the sequence {fn } is a K-frame for H with respect to H0 , if there exist constants A, B > 0 such that ∑ | ⟨h, fn ⟩ |2 ≤ B ∥h∥2 , h ∈ H. A ∥K ∗ h∥2 ≤ n Definition 1.6. Let F : X → H be weakly measurable. We define the ∫ map X ·F dµ : L2 (X) → H as follows: ⟩ ⟨∫ ∫ gF dµ, h := g (x) ⟨F (x) , h⟩ dµ, h ∈ H, g ∈ L2 (X) . X X ∫ It is clear ∫ that, the vector valued integral X gF dµ exists in H if for each h ∈ H, X g (x) ⟨F (x), h⟩ dµ exists. Definition 1.7. Let H0 ⊆ H. Suppose that F : X → H is weakly measurable and K ∈ B (H0 , H). Then F is called a family of local cK-atoms for H0 if the following conditions are satisfied: ∫ (i) For each g ∈ L2 (X) the vector valued integral X gF dµ exists in H. (ii) There exist some a > 0 and ℓ : X → L(H0 , C) such that for each h ∈ H0 , ℓ (·) (h) ∈ L2 (X) and also ∫ ℓ (·) (h) F dµ. ∥ℓ(·)(h)∥2 ≤ a ∥h∥ , Kh = X If K is the identity function on H0 then F is called a family of local atoms for H0 . 172 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI Definition 1.8. Let K ∈ B (H0 , H) and F : X → H be weakly measurable. Then the map F is called a cK-frame with respect to H0 , if there exist constants A, B > 0 such that for each h ∈ H, ∫ 2 ∗ |⟨F (x) , h⟩|2 dµ ≤ B ∥h∥2 . A ∥K h∥ ≤ X A cK-frame F is called a Parseval cK-frame, whenever for every h ∈ H, ∫ |⟨F (x) , h⟩|2 dµ = ∥K ∗ h∥2 . X Lemma 1.9 ([13]). Let F∫ : X → H be weakly measurable. For each φ ∈ L2 (X) , the value of X φF dµ exists in H if and only if for each h ∈ H, ⟨F, h⟩ ∈ L2 (X). Lemma 1.10 ([12]). Let F : X → H be weakly measurable. Then F is ∫ a c-Bessel mapping for H if and only if for each φ ∈ L2 (X), X φF dµ exists in H. Remark 1.11. Let F : X → H be a c-Bessel mapping for H. The synthesis operator is defined by ∫ 2 φF dµ. TF : L (X) → H, TF (φ) = X Hence, for each φ ∈ L2 (X) and h ∈ H, ⟩ ∫ ⟨∫ φ (x) ⟨F (x) , h⟩ dµ. φF dµ, h = X X The analysis operator is defined by TF∗ : H → L2 (X) , TF∗ (h) = ⟨h, F ⟩ . So, for the frame operator SF := TF TF∗ we have ∫ ⟨h, F ⟩ F dµ, h ∈ H. SF (h) = X Theorem 1.12 ([12]). Let H0 ⊆ H. Let F : X → H be weakly measurable, and K ∈ B (H0 , H). Then the following assertions are equivalent: (i) F is a family of local cK-atoms for H0 . (ii) F is a cK-frame for H with respect to H0 . ( ) (iii) F is a c-Bessel mapping for H, and there exists G ∈ B H0 , L2 (X) such that ∫ G (h) F dµ, h ∈ H0 . Kh = X SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 173 Theorem 1.13 ([12]). Let K ∈ B (H0 , H), and F : X → H be a cKframe for H with respect to H0 , with bounds A,B. If K is closed range then SF is invertibale on R (K), and for each h ∈ R (K) (1.1) A K† −2 ∥h∥2 ≤ ⟨SF (h) , h⟩ ≤ B ∥h∥2 . 2. Operators Preserving ck-Frames Theorem 2.1. Suppose that F : X → H is a cK-frame for H and U ∈ B(H) with R(U ) ⊆ R(K). Then F is a cU -frame for H. Proof. Let F be a cK-frame for H with bounds A and B. Since R(U ) ⊆ R(K), by Theorem 1.4 there exists α > 0 such that U U ∗ ≤ α2 KK ∗ . By the definition of cK-frames, for each h ∈ H we have Aα−2 ∥U ∗ (h)∥2 ≤ A ∥K ∗ (h)∥2 ∫ ≤ |⟨h, F (x)⟩|2 dµ. X Hence, F is a cU-frame for H. □ Theorem 2.2. Let K ∈ B(H) with dense range, F : X → H be a cKframe and U ∈ B(H) be closed range. If U F is a cK-frame for H then U is surjective. Proof. suppose U F is a cK-frame for H with frame bounds A and B. Then for any h ∈ H we have ∫ 2 ∗ (2.1) A ∥K h∥ ≤ |⟨h, U F (x)⟩|2 dµ ≤ B ∥h∥2 . X Since K is with dense range, K ∗ is injective. By (2.1), N (U ∗ ) ⊂ N (K ∗ ), then U ∗ is injective. Moreover R (U ) = N (U ∗ )⊥ = H. Thus, U is surjective. □ Theorem 2.3. Suppose K ∈ B (H) and let F : X → H be a cK-frame for H. If U ∈ B (H) has closed range with U K = KU , then U F : X → H is a cK-frame for R (U ). Proof. Since U has closed range, then the pseudo-inverse U † such ( † )∗it has ∗ † ∗ U . Then for each h ∈ R (U ), that U U = I. Now I = I = U ( † )∗ ∗ ∗ ∗ K h= U U K h. So we have ∥K ∗ h∥ = (U † )∗ U ∗ K ∗ h ≤ (U † )∗ ∥U ∗ K ∗ h∥ . 174 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI −1 Therefore, (U † )∗ ∥K ∗ h∥ ≤ ∥U ∗ K ∗ h∥. Now for each h ∈ R(U ), ∫ ∫ 2 |⟨h, U F (x)⟩| dµ = |⟨U ∗ h, F (x)⟩|2 dµ X X ≥ A ∥K ∗ U ∗ h∥2 = A ∥U ∗ K ∗ h∥2 ≥ A (U † )∗ −2 ∥K ∗ h∥2 . Since F is a c-Bessel mapping with bound B, we have ∫ ∫ |⟨h, U F (x)⟩|2 dµ = |⟨U ∗ h, F (x)⟩|2 dµ X X ≤ B ∥U ∗ h∥2 ≤ B ∥U ∥2 ∥h∥2 . Therefore, U F is a cK-frame for R(U ). □ Remark 2.4. From Theorems 2.2 and 2.3 we conclude the following: Let K ∈ B(H) be with dense range. Let F be a cK-frame for H and U ∈ B(H) has closed range with U K = KU . Then U F is a cK-frame for H if and only if U is surjective. Theorem 2.5. Suppose K ∈ B (H) has dense range, F is a cK-frame and U ∈ B (H) has closed range. If U F and U ∗ F are cK-frames for H, then U is invertible. Proof. Suppose U F is a cK-frame for H with frame bounds A1 and B1 . Then for any h ∈ H ∫ 2 ∗ |⟨h, U F (x)⟩|2 dµ ≤ B1 ∥h∥2 . (2.2) A1 ∥K h∥ ≤ X Since K has dense range, then K ∗ is injective. By (2.2) we have N (U ∗ ) ⊂ N (K ∗ ), therefore U ∗ is injective. Moreover R(U )=N (U ∗ )⊥ = H, then U is surjective. Suppose A2 and B2 are frame bounds for U ∗ F , then for any h ∈ H, ∫ 2 ∗ |⟨h, U ∗ F (x)⟩|2 dµ ≤ B2 ∥h∥2 . (2.3) A2 ∥K h∥ ≤ X K∗ As K has dense range, is injective. Then, by (2.3) we get N (U ) ⊂ N (K ∗ ), so U is injective. Thus U is bijective. Now, by the Bounded Inverse Theorem, U is invertible. □ Theorem 2.6. Let K ∈ B(H) and F be a cK-frame for H and U ∈ B(H) be a co-isometry with U K = KU . Then U F is a cK-frame for H. SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 175 Proof. Let F be a cK-frame for H. Since U is a co-isometry, we have for each h ∈ H ∫ ∫ |⟨h, U F (x)⟩|2 dµ = |⟨U ∗ h, F (x)⟩|2 dµ X X ≥ A ∥K ∗ u∗ h∥2 = A ∥U ∗ K ∗ h∥2 = A ∥K ∗ h∥2 . It is clear that U F is a c-Bessel mapping. Since F : X → H is a c-Bessel mapping, then for each h ∈ H ∫ ∫ |⟨U ∗ h, F (x)⟩|2 dµ |⟨h, U F (x)⟩|2 dµ = X X ≤ B ∥U ∥2 ∥h∥2 . Therefore, U F is a cK-frame for H. □ Theorem 2.7. Let F : X → H be a c-Bessel mapping for H. Then F : X → H is a cK-frame for H if and only if there exists A > 0 such that SF ≥ AKK ∗ , where SF is the frame operator for F. Proof. F : X → H is a cK-frame for H with frame bounds A, B and frame operator SF , if and only if, ∫ 2 ∗ |⟨h, F (X)⟩|2 dµ A ∥K h∥ ≤ X = ⟨SF (h), h⟩ ≤ B ∥h∥2 , if and only if, ∀h ∈ H, ⟨AKK ∗ h, h⟩ ≤ ⟨SF (h), h⟩ ≤ ⟨Bh, h⟩ , if and only if, ∀h ∈ H, SF ≥ AKK ∗ . □ Theorem 2.8. Let F : X → H be a c-frame for H. Then KF : X → H and K ∈ B(H) is a cK-frame for H. Proof. By the definition of c-frame we have ∫ ∫ 2 |⟨K ∗ h, F (x)⟩|2 dµ |⟨h, KF (x)⟩| dµ = X X ≤ B ∥K ∗ h∥2 ≤ B ∥K ∗ ∥2 ∥h∥2 . 176 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI So, KF is a Bessel mapping. By theorem 1.12 it is sufficient to show that KF is∫an atomic system for H. For each h ∈ H we have ⟨h, KF ⟩ ∈ L2 (X), so X g (KF ) dµ ∈ H for each g ∈ L2 (X). By Theorem 3.5 in ⟨ ⟩ [8], for each h ∈ H we have h = TF ( SF−1 (h), F ); therefore ⟨ ⟩ Kh = KTF ( SF−1 (h), F ) ⟨ ⟩ = TKF ( SF−1 (h), F ) ∫ ⟨ −1 ⟩ SF (h), F (x) KF (x) dµ. = X So, for all h1 ∈ H ⟨Kh, h1 ⟩ = = Let ⟨∫ ∫ X X ⟨ ⟨ SF−1 (h), F (x) ⟩ KF (x) dµ, h1 ⟩ ⟩ SF−1 (h), F (x) ⟨h1 , KF (x)⟩ dµ. ℓ : X → L(H, C), ⟨ ⟩ ℓ(x)(h) = SF−1 (h), F (x) , h ∈ H, x ∈ X. So, for each h ∈ H and x ∈ X, we get ℓ(x)(h) ∈ L2 (X) and (∫ )1 2 ⟨ −1 ⟩2 ∥ℓ(x)(h)∥2 = dµ SF (h), F (x) X Now, if a := √ ( )1 2 2 ≤ B SF−1 (h) √ ≤ B SF−1 ∥h∥ . B SF−1 , by Definition 1.7 the proof is completed. □ 3. Some Identities and Inequalities for cK-Frames In this section, we introduce some useful identities and inequalities by frame operators. Let K ∈ B(H0 , H), F : X → H be c-Bessel mappings for H and G : X → H0 be a c-Bessel mapping for H0 . We say that F , G is a cK-dual pair, if Kh0 = TF (⟨h0 , G⟩) , for any h ∈ H and h0 ∈ H0 . In this case, we know that F is a cK-frame for H with respect to H0 and G is a cK ∗ -frame for H0 with respect to H (for more details, we refer to [12]). Now, we define ∫ ⟨h, G(x)⟩ F (x) dµ, MX1 h := X1 SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 177 for each h ∈ H. So, MX1 h is well-dfined and bounded. Indeed, if h ∈ H then ( )2 ⟨ ⟩ 2 ′ ∥MX1 h∥ = sup MX1 h, h ∥h′ ∥=1 = ≤ ( ∫ ⟨∫ sup ∥h′ ∥=1 X1 ⟨h, G(x)⟩ F (x) dµ, h′ 2 X1 |⟨h, G(x)⟩| dµ. sup ∥h′ ∥=1 ≤ BB ′ ∥h∥2 , ∫ X1 ⟨ ⟩ )2 F (x), h′ ⟩ 2 dµ where, B, B ′ are uper bounds for F, G, respectively. It is easy to check that MX1 + MX1c = K where, X1c is the complement of X1 . Theorem 3.1. Let F be a cK-frame for H with the dual G. Then for each measureable subspace X1 ⊆ X and h ∈ H, ∫ ⟨h, G(x)⟩ ⟨Kh, F (x)⟩dµ − ∥MX1 h∥2 X1 ∫ 2 ⟨h, G(x)⟩ ⟨Kh, F (x)⟩ dµ − MX1c h . = X1c Proof. Suppose that h ∈ H and X1 ⊆ X. We have ∫ 2 X1 ⟨h, G(x)⟩ ⟨Kh, F (x)⟩dµ − ∥MX1 h∥ = ⟨MX1 h, Kh⟩ − ⟨MX1 h, MX1 h⟩ ⟩ ⟨ ∗ MX1 h, h = ⟨K ∗ MX1 h, h⟩ − MX 1 ⟩ ⟨ ∗ )MX1 h, h = (K ∗ − MX 1 ⟨ ⟩ ∗ = MX c (K − MX c )h, h 1 1 ⟨ ⟩ ⟨ ⟩ ∗ ∗ = MX − M c Kh, h c MX c h, h X 1 1 1 ⟨ ⟩ 2 ∗ = h, K MX1c h − MX1c h ∫ = ⟨h, G(x)⟩ ⟨Kh, F (x)⟩ dµ − MX1c h 2 . X1c □ Theorem 3.2. Let F : X → H be a Parseval cK-frame for H. For every h ∈ H, X1 ⊆ X and E ⊆ X1c we have ∫ 2 X1 ∪E ⟨h, F (x)⟩ F (x) dµ − ∫ 2 X1c |E ⟨h, F (x)⟩ F (x) dµ 178 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI ∫ = 2 − ⟨h, F (x)⟩F (x) dµ X1 + 2Re ∫ E ∫ 2 X1c ⟨h, F (x)⟩ F (x) dµ ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ. Proof. For each measurable subspace X1 ⊆ X, we define ∫ ⟨h, F (x)⟩ F (x) dµ. SX1 h = X1 c =KK ∗ . We have SX1 +SX1 Therefore, 2 2 2 2 ∗ c = SX − (KK − SX1 ) SX − SX 1 1 1 = KK ∗ SX1 + SX1 KK ∗ − (KK ∗ )2 = KK ∗ SX1 − (KK ∗ − SX1 ) KK ∗ c KK ∗ . = KK ∗ SX1 − SX 1 Hence, for every h ∈ H we obtain ∫ 2 X1 ∪E ⟨h, F (x)⟩ F (x) dµ − ∫ X1c |E 2 ⟨h, F (x)⟩ F (x) dµ ⟨ ⟩ ⟨ ⟩ = KK ∗ SX1 ∪ E h, h − SX1c |E KK ∗ h, h ⟨ ⟩ ⟨ ⟩ = SX1 ∪ E h, KK ∗ h − KK ∗ h, SX1c |E h ⟩ ⟨∫ ⟨ ⟩ ∗ ⟨h, F (x)⟩ F (x) dµ, KK h − SX1c |E h, KK ∗ h = ∪ X1 = ∫ X1 − = ∫ ∫ − ∫ + ∫ ∫ ∪ E ⟨h, F (x)⟩⟨F (x), KK ∗ h⟩ dµ X1c |E X1 + = ∫ E ⟨h, F (x)⟩ ⟨F (x), KK ∗ h⟩ dµ ⟨h, F (x)⟩ ⟨F (x), KK ∗ h⟩ d µ E ⟨h, F (x)⟩ ⟨F (x), KK ∗ h⟩ dµ X1c E X1 ⟨h, F (x)⟩ ⟨F (x), KK ∗ h⟩ dµ ⟨h, F (x)⟩ ⟨F (x), KK ∗ h⟩ dµ 2 ⟨h, F (x)⟩ F (x)dµ − ∫ 2 X1c ⟨h, F (x)⟩ F (x) dµ SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 179 + 2Re ∫ E ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ. □ Theorem 3.3. Let F : X → H be a Parseval cK-frame for H. For every h ∈ H and X1 ⊆ X we have, (∫ ) ∫ 2 ∗ Re ⟨h, F (x)⟩ ⟨KK h, F (x)⟩ dµ + ⟨h, F (x)⟩ F (x) dµ X1c X1 = Re (∫ + ∫ ≥ X1 X1c ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ 2 ) ⟨h, F (x)⟩ F (x) dµ 3 ∥KK ∗ h∥2 . 4 ∗ ∗ 2 − S 2 = KK ∗ S Proof. Since SX X1 − SX1c KK and SX1 + SX1c = KK , X1c 1 we can write ( )2 KK ∗ (KK ∗ )2 2 2 SX1 + SX1c = 2 − S X1 + 2 2 ≥ (KK ∗ )2 . 2 Consequently ( )∗ 2 ∗ 2 2 ∗ 2 c + SX1 KK + SX c = KK ∗ SX1 + SX KK ∗ SX1 + SX1 c + KK SX1 + SX c 1 1 1 ( ) ∗ 2 2 = KK SX1 + SX1c + SX1 + SX1c 2 2 c + SX = (SX1 + SX1c )KK ∗ + SX 1 1 ≥ 3 (KK ∗ )2 . 2 Thus, we obtain ) (∫ ∫ ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ + Re X1c = Re (∫ X1 2 X1 ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ ) ⟨h, F (x)⟩ F (x) dµ + ∫ 2 X1c ⟨h, F (x)⟩ F (x) dµ ⟨ ⟩ ⟨ ⟩) 1( 2 ∗ 2 c h, h c ⟨KK ∗ SX1 h, h⟩ + SX + ⟨h, KK S h⟩ + h, S h X1 X1 1 2 3 ≥ ∥KK ∗ h∥2 . 4 = 180 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI □ Theorem 3.4. Let K be a closed operator and F : X → H be a cKframe for H with the optimal lower bound A. Then, (I) For each h ∈ H, ∫ ∫ 2 |⟨h, F (x)⟩|2 d µ. ⟨h, F (x)⟩ F (x) dµ ≤ ∥SF ∥ X X (II) For any h ∈ R(K), ∫ 1 |⟨h, F (x)⟩|2 dµ ≤ K† A X 2 ∫ 2 X ⟨h, F (x)⟩ F (x) dµ . (I) For any h ∈ H, we can write ∫ 2 ⟨h, F (x)⟩ F (x) dµ = |⟨SF (h), SF (h)⟩| Proof. X ≤ ∥SF ∥ |⟨SF (h), h⟩| ∫ |⟨h, F (x)⟩|2 dµ. = ∥SF ∥ X (II) For each h ∈ R(K), )2 (∫ 2 |⟨h, F (x)⟩| dµ ≤ |⟨SF (h), h⟩|2 X ≤ ∥SF (h)∥2 ∥h∥2 ( )∗ ≤ ∥SF (h)∥2 K † K ∗ h 1 ∥SF (h)∥2 A and the proof is completed. 2 ∥K ∗ h∥2 ∫ 2 † |⟨h, F (x)⟩|2 dµ, (K ) ≤ ∥SF (h)∥2 (K † ) ≤ 2 X □ Some applications of above theorems, we present the following interesting assertions. Let F : X → H be a cK-Parseval frame for H. We consider Re (∫ X1c Re (∫ 1 υ+ (F, K, X1 ) := sup h̸=0 υ− (F, K, X1 ) := inf h̸=0 ) ∫ ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ + X1 ⟨h, F (x)⟩ F (x) dµ ∥KK ∗ h∥2 ) ∫ ∗ ⟨h, F (x)⟩ ⟨KK h, F (x)⟩ dµ + X1 ⟨h, F (x)⟩ F (x) dµ c X ∥KK ∗ h∥2 Theorem 3.5. Suppose that F : X → H is a cK-Parseval frame for H. The following assertions hold: 2 , 2 . SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 181 (I) ) ( 3 ≤ υ− (F, K, X1 ) ≤ υ+ (F, K, X1 ) ≤ ∥K∥ K † 1 + ∥K∥ K † . 4 (II) υ+ (F, K, X1 ) = υ+ (F, K, X1c ) and υ− (F, K, X1 ) = υ− (F, K, X1c ). Proof. (I). It is enough to prove the upper inequality. By Theorem 3.4, (I), we get ∫ ∫ 2 |⟨h, F (x)⟩|2 dµ ⟨h, F (x)⟩ F (x) dµ ≤ ∥SX1 ∥ X1 ∫X 1 |⟨h, F (x)⟩|2 dµ ≤ ∥SX1 ∥ X ∗ ≤ ∥K∥ ∥K h∥2 2 = ∥K∥2 KK † K ∗ h ≤ ∥K∥2 K † 2 2 ∥KK ∗ h∥2 . Moreover, Re (∫ ∗ X1c ⟨h, F (x)⟩ ⟨KK h, F (x)⟩ dµ ≤ (∫ 2 X ∗ |⟨h, F (x)⟩| dµ = ∥K h∥ ∥K ∗ KK ∗ h∥ ) ) 12 (∫ 2 ∗ X |⟨KK h, F (x)⟩| dµ ) 21 = KK † K ∗ h ∥K ∗ KK ∗ h∥ 2 ≤ ∥K∥ K † ∥KK ∗ h∥ . Therefore, υ− (F, K, X1 ) ≤ υ+ (F, K, X1 ) ≤ ∥K∥ K † (II). By the proof of Theorem 3.2, we have ( ) 1 + ∥K∥ K † . 2 2 c. SX + SX!c KK ∗ = KK ∗ SX1 + SX 1 ! Hence, for any h ∈ H, ⟩ ⟩ ⟨ 2 ⟩ ⟨ ⟨ 2 c h, h + ⟨KK ∗ SX1 h, h⟩ . SX1 h, h + SX1c KK ∗ h, h = SX 1 So, ∫ 2 X1 ⟨h, F (x)⟩ F (x) dµ = ∫ X1c + ∫ X1c ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ ⟨h, F (x)⟩ F (x) dµ 2 + ∫ X1 ⟨h, F (x)⟩ ⟨KK ∗ h, F (x)⟩ dµ, 182 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI and this results (II). □ 4. Perturbation of cK-Frames Throughout this section, the orthogonal projection of H onto a closed subspace V ⊆ H is denoted by ΠV . Theorem 4.1. Let F : X → H be a cK frame for H with bounds A, B, and µ be a σ-finite measure. Let G : X → H be weakly measurable { and assume that }there exist constants λ1 , λ2 , γ ≥ 0 such that γ max λ1 + √ K † , λ2 < 1 and A (4.1) ∫ φ(x) ⟨F (x) − G(x), h⟩ dµ X ∫ ∫ φ(x) ⟨G(x), h⟩ dµ + γ ∥φ∥2 , φ(x) ⟨F (x), h⟩ dµ + +λ2 ≤ λ1 X L2 (X) X for each φ ∈ and h ∈ (H)1 . Then G : X → H is a continuous ΠQ(R(k)) K- frame for H with bounds [√ ]2 ]2 [√ −1 (1 − λ1 ) − γ A K† B(1 + λ1 ) + γ , , (1 + λ2 )2 ∥K∥2 (1 − λ2 )2 where Q = UG TF∗ and TF , UG are synthesis operators for F and G, respectively. Proof. The condition (4.1) implies that for all φ ∈ L2 (X) and h ∈ (H)1 ∫ φ (x) ⟨G(x), h⟩ dµ X ∫ ∫ φ (x) ⟨F (x), h⟩ dµ φ (x) ⟨F (x) − G(x), h⟩ dµ + ≤ X X ∫ ∫ φ (x) ⟨G(x), h⟩ dµ + γ ∥φ∥2 . φ (x) ⟨F (x), h⟩ dµ + λ2 ≤ (1 + λ1 ) X So ∫ X φ (x) ⟨G(x), h⟩ dµ ∫ 1 + λ1 γ ≤ ∥φ∥2 φ (x) ⟨F (x), h⟩ dµ + 1 − λ2 X 1 − λ2 [( ∫ )1 ] ) 1 (∫ 2 2 1 + λ1 γ 2 2 |⟨F (x), h⟩| dµ ≤ + |φ (x)| dµ ∥φ∥2 1 − λ2 1 − λ2 X X X SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 183 ≤ Let ( 1 + λ1 √ γ B+ 1 − λ2 1 − λ2 ) ∥φ∥2 . UG :L2 (x) → H, ∫ φ (x) ⟨G(x), h⟩ dµ, ⟨(UG )φ, h⟩ = X for all h ∈ H and φ ∈ L2 (X). Then ∥(UG ) φ∥ = sup |⟨(UG )φ, h⟩| h∈(H)1 = sup h∈(H)1 ≤ ( ∫ X φ(x) ⟨G(x), h⟩ dµ 1 + λ1 √ γ B+ 1 − λ2 1 − λ2 ) ∥φ∥2 . Thus, UG is bounded, so G is a c-Bessel mapping for H with bound [√ ]2 B (1 + λ1 ) + γ . Now, we prove that G has a lower cK-frame (1 − λ2 )2 bound. By Remark 1.11 we can define the following operators for all φ ∈ L2 (X) TF : L2 (X) ∫ → H, TF (φ) = φF dµ, By (4.1) we obtain (4.2) UG : L2 (X) ∫ → H, UG (φ) = φGdµ. |⟨TF (φ) − UG (φ) , h⟩| ≤ λ1 |⟨TF (φ) , h⟩| + λ2 |⟨UG (φ) , h⟩| + γ ∥φ∥2 . Now, let TF∗ (h′ ) := φ , h′ ∈ R (K). By (4.2) we have (4.3) |⟨SF (h′ ) − UG TF∗ (h′ ), h⟩| ≤ λ1 |⟨SF (h′ ), h⟩| + λ2 |⟨UG TF∗ (h′ ), h⟩| + γ ∥TF∗ (h′ )∥l2 , for any h′ ∈ R (K). By (1.1), we get ⟩ ⟨ 2 TF∗ (h′ ) = SF (h′ ), h′ ≤ SF (h′ ) ≤ A−1 K † h′ 2 SF (h′ ) 2 SF (h′ ) 2 , thus (4.4) TF∗ (h′ ) 2 ≤ A−1 K † 2 , for each h′ ∈ R (K). By (4.3) and (4.4), for any h′ ∈ R (K), we have ⟨ ⟩ SF (h′ ) − UG TF∗ (h′ ), h) 184 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI ⟨ ≤ λ1 So, SF (h′ ), h ⟩ ⟨ + λ2 UG TF∗ (h′ ), h γ +√ K† SF (h′ ) A γ K † SF (h′ ) ∥h∥ + √ A ⟩ ≤ λ1 SF (h′ ) ∥h∥ + λ2 UG TF∗ (h′ ) ( ) γ † K = λ1 ∥h∥ + √ SF (h′ ) + λ2 UG TF∗ (h′ ) A SF (h′ ) − UG TF∗ (h′ ) = sup h∈(H)1 ≤ ( ⟨ SF (h′ ) − UG TF∗ (h′ ), h γ K† λ1 + √ A ) ∥h∥ . ⟩ SF (h′ ) + λ2 UG TF∗ (h′ ) . Therefore, we can write ) ( γ † K 1 − λ1 + √ A SF (h′ ) ≤ UG TF∗ (h′ ) , (4.5) 1 + λ2 and γ 1 + λ1 + √ K† A UG TF∗ (h′ ) ≤ 1 − λ2 (4.6) SF (h′ ) . Combinig (1.1), (4.5) and (4.6), we have (4.7) [ )] ( γ −2 A K† K† 1 − λ1 + √ A ∥h′ ∥ ≤ ∥UG TF∗ (h′ )∥ 1 + λ2 [ ] γ 1 + λ1 + √ K† B A ≤ ∥h′ ∥ , 1 − λ2 for each h′ ∈ R (K). Let Q := UG TF∗ . Now, we prove that R (Q) is closed. In fact, for each {yn }∞ n=1 ⊂ R (Q) with lim yn = y, n→∞ y ∈ H, there exists xn ∈ R (K) such that (4.8) yn = Q (xn ) . By (4.7) and (4.8) we have ∥xn − xm ∥ ≤ D−1 ∥Q(xn − xm )∥ ≤ D−1 ∥ym − yn ∥ , SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 185 where )] γ −2 † A K† K 1 − λ1 + √ A D= . 1 + λ2 It follows that {xn }∞ n=1 is a cauchy sequence, so there exists x ∈ R (K) such that limn→∞ xn = x. By the continuity of Q we have [ ( y = lim yn = lim Q (xn ) = Q (x) ∈ R (Q) , n→∞ n→∞ which implies that R (Q) is closed. By (4.7), we know that Q is injective on R (K). Then, we conclude that Q : R (K) → R (Q) is invertible. Thus, combining with (4.5) and (4.7) we obtain that, for all y ∈ Q (R (K)) 1 + λ2 ) ∥y∥ , γ † 1 − λ1 + √ ∥K ∥ A SF Q−1 (y) ≤ (4.9) Q−1 (y) ≤ [ (4.10) ( 1 + λ2 )] ∥y∥ . γ −2 1 − λ1 + √ ∥K † ∥ A ∥K † ∥ A ( Now, for each h1 ∈ H, we get ΠQ(R(k)) Kh1 = QQ−1 ΠQ(R(K)) Kh1 ) ( = UG TF∗ Q−1 ΠQ(R(k)) Kh1 ∫ ( ∗ −1 ) TF Q ΠQ(R(k)) Kh1 G dµ = ∫X ψG dµ, = X where ψ := TF∗ (Q−1 ΠQ(R(K)) Kh1 ) K ∗ (ΠQ(R(K)) )∗ h ⟨ ∗ ⟩ = sup K (ΠQ(R(K)) )∗ h, h1 h1 ∈(H)1 = sup h1 ∈(H)1 = sup h1 ∈(H)1 = sup h1 ∈(H)1 = sup h1 ∈(H)1 ⟨ ⟨ h, (ΠQ(R(K)) )Kh1 ⟩ (ΠQ(R(K)) )Kh1 , h ⟨∫ ∫ X ψG dµ, h X ⟩ ⟩ ψ(x) ⟨G(x), h⟩ dµ ∈ L2 (X). Hence, for any h ∈ H 186 GH. RAHIMLOU, R. AHMADI, M. A. JAFARIZADEH, AND S. NAMI ≤ = sup h1 ∈(H)1 (∫ sup ∥ψ∥2 h1 ∈(H)1 = sup sup h1 ∈(H)1 ≤ SF Q X |ψ(x)| dµ (∫ ) 12 (∫ 2 X 2 X |⟨G(x), h⟩| dµ TF∗ Q−1 ΠQ(R(K)) Kh1 2 h1 ∈(H)1 = 2 (⟨ −1 |⟨G(x), h⟩| dµ ) 12 ) 12 (∫ 2 X |⟨G(x), h⟩| dµ SF Q−1 ΠQ(R(K)) Kh1 , Q−1 ΠQ(R(K)) Kh1 ΠQ(R(K)) K 1 2 −1 Q ΠQ(R(K)) K 1 2 (∫ ) 21 ⟩) 12 (∫ 2 X |⟨G(x), h⟩| dµ 2 X |⟨G(x), h⟩| dµ ) 21 (∫ ) 21 (1 + λ2 ) 2 )√ ] ( ≤[ ∥K∥ |⟨G(x), h⟩| dµ −1 X 1 − λ1 + √γA ∥K † ∥ A ∥K † ∥ ) 12 (∫ (1 + λ2 ) ∥K∥ 2 ] |⟨G(x), h⟩| dµ . = [√ −1 X A ∥K † ∥ (1 − λ1 ) − γ Thus, for each h ∈ H ]2 [√ −1 (1 − λ1 ) − γ A K† 2 2 (1 + λ2 ) ∥K∥ K ∗ (Π∗Q(R(K)) )h 2 ≤ ∫ X |⟨G(x), h⟩|2 dµ. □ References 1. S.T. Ali, J.P. Antoine, and J.P. Gazeau, Continuous frames in Hilbert spaces, Ann.Phys., 222 (1993), pp. 1-37. 2. H. Bolcskel, F. Hlawatsch, and H.G. Feichyinger, Frame-Theoretic analysis of oversampled filter bank, IEEE Trans. Signal Processing., 46 (1998), pp. 3256- 3268. 3. P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and Distributed Processing, Appl. Comput. Harmon. Anal., 25 (2008), pp. 114-132. 4. O. Christensen, Introduction to frames and Riesz bases, Birkhäuser, Boston, 2003. 5. I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal Expansions, J. Math. Phys., 27 (1986), pp. 1271-1283. 6. R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert spaces, Proc. Amer. Math. Soc., 17 (1966), pp. 413-415. ) 21 SOME PROPERTIES OF CONTINUOUS K-FRAMES IN HILBERT SPACES 187 7. R.J. Duffin and A.C. Schaeffer, A class of nonharmonik Fourier series, Trans. Amer. Math. Soc., 72 (1952), pp. 341-366. 8. M.H. Faroughi and E. Osgooei, C-Frames and C-Bessel Mappings, Bull. Iran. Math., 38 (2012), pp. 203-222. 9. L. Găvruţa, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012), pp. 139-144. 10. G. Kaiser, A Friendly Guide to Wavelets, Birkhäuser, Boston, 1994. 11. A. Rahimi, A. Najati, and Y.N. Dehgan, Continuous frame in Hilbert space, Methods of Functional Analysis and Topology., 12 (2006), pp. 170-182. 12. GH. Rahimlou, R. Ahmadi, M.A. Jafarizadeh, and S. Nami, Continuous k-Frames and their duals, (2018) Submitted. 13. W. Rudin, Real and Complex Analysis, New York, Tata Mc GrawHill Editions, 1987. 14. X. Xiao, Y. Zhu, and L. Găvruţa, Some Properties of K-frames in Hilbert Spaces, Results. Math., 63 (2012), pp. 1243-1255. 1 Department of Mathematics, Shabestar Branch, Islamic Azad University, Shabestar, Iran. E-mail address: [email protected] 2 Institute of Fundamental Sciences, University of Tabriz, Tabriz, Iran. E-mail address: [email protected] 3 Faculty of Physic, University of Tabriz, Tabriz, Iran. E-mail address: [email protected] 4 Faculty of Physic, University of Tabriz, Tabriz, Iran. E-mail address: [email protected]